A preliminary study of endothelial cell migration during angiogenesis using a meshless method approach

Abstract

Angiogenesis, the development of new blood capillaries, is crucial for the wound healing process. This biological process allows the proper blood supply to the tissue, essential for cell proliferation and viability. Several biological factors modulate angiogenesis, however the vascular endothelial growth factor (VEGF) is the main one. Given the complexity of angiogenesis, in the last years, computational modelling aroused the interest of scientists since it allows to model this process with different, more economic and faster methodologies, comparatively to experimental approaches. In this work, a mathematical model motivated by the analysis of the effect of VEGF diffusion gradient in endothelial cell migration is presented. This is the process that allows capillary formation and it is essential for angiogenesis. The proposed mathematical model is combined with the Radial Point Interpolation Method, being the area discretized considering an unorganized nodal cloud and a background mesh of integration points, without predefined relations. The nodal connectivity was achieved using the “influence‐domain” approach. The interpolation functions were constructed using the Radial Point Interpolators techniques. This method combines a radial basis functions with a polynomial functions to obtain the approximation. This preliminary work does not account for the whole complexity of cell and tissue biology, and numerical results are presented for an idealised two‐dimensional setting. Nevertheless, the developed RPIM software is a valid numerical tool that can be adjusted to biological problems and may also be able to complement the biological and medical subjects.

Computational Insights into the Interplay of Mechanical Forces in Angiogenesis

“…In this study, the discretisation method used was the Radial Point Interpolation Method (RPIM) [28,29]. RPIM has previously proven its efficacy in modelling angiogenesis [16,30], and the comprehensive RPIM formulation can be found in the literature [31].…”

“…Consequently, m = 3. An additional equation system must be included to obtain an unique solution [30]:…”

“…Modelling endothelial cell migration during angiogenesis, as described in previous studies [16,30], entailed accounting for the chemical diffusion of VEGF within a homogeneous medium. Additionally, numerical simulation of VEGF diffusion as a field problem is feasible [33].…”

“…The simulation of capillary network growth was conducted within 5 × 5 mm 2 square domain, discretised with 2601 nodes to create a regular nodal mesh. In this study, three dif-ferent initial domain geometries were defined, as illustrated in Figure 1, aligning with approaches from previous works [16,30,34]. In the performed simulations, the location of the capillary lumen, the endothelial cell monolayer and the VEGF release region change in each model under analysis.…”

Role of Angiogenesis and Its Biomarkers in Development of Targeted Tumor Therapies

“…VEGF induces ECs proliferation via the ERK (extracellular signal-regulated kinase) and PI3K/Akt (phosphoinositide 3-kinases/protein kinases B) pathways. ECs migration downstream of VEGFR2 is induced through signalling pathway involving Rho GTPases and PI3K activation [ 73 ]. VEGF overexpression has been reported in solid tumors, therefore VEGF is considered as a potential marker for cancer [ 74 ].VEGF-A, angiogenic multifunctional mediator, binds to extracellular domain of VEGFR2 and transduce the responses of VEGF in ECs including ECs survival and proliferation, migration, permeability, and formation of capillary lumen, thus orchestrating the vasculature of cancer.…”